Topic 12 · page 11 of 17

Building a Born-Haber cycle

A Born-Haber cycle is Hess's law for an ionic solid. One route goes straight from the elements to the solid. The other goes through gaseous atoms and gaseous ions, and ends with the lattice energy.

12 min read12.13

Read first: Three definitions for ionic solids

Two routes, one total 12.13

The lattice energy of sodium chloride cannot be measured directly. You cannot collect separate gaseous Na⁺ and Cl⁻ ions and watch them form a crystal. But the enthalpy change of formation of NaCl can be measured, and so can every other step on a longer route. Hess's law (Topic 6) says that two routes between the same start and finish have the same total enthalpy change. Put the unknown on one route and everything else on the other.

The steps, in order 12.13

For sodium chloride, route 2 has five steps. The first four have values you can look up. The fifth is the unknown.

Step What happens Equation ΔH\Delta H / kJ mol⁻¹
1 atomise sodium Na(s)→Na(g)\ce{Na(s) -> Na(g)} +107
2 atomise chlorine 12 ClX2(g)→Cl(g)\ce{1/2 Cl2(g) -> Cl(g)} +122
3 ionise sodium (first ionisation energy) Na(g)→NaX+(g)+eX−\ce{Na(g) -> Na+(g) + e-} +496
4 add the electron to chlorine (first electron affinity) Cl(g)+eX−→ClX−(g)\ce{Cl(g) + e- -> Cl-(g)} −349
5 form the lattice (lattice energy) NaX+(g)+ClX−(g)→NaCl(s)\ce{Na+(g) + Cl-(g) -> NaCl(s)} unknown

Draw each species as a horizontal level, higher for more enthalpy. Endothermic steps go up and exothermic steps go down. The figure is drawn to scale, with the elements at zero.

An enthalpy level diagram for sodium chloride, drawn to scale. Starting at zero for sodium solid plus half a chlorine molecule, arrows go up by 107 to Na gas, up by 122 to Na gas plus Cl gas, up by 496 to Na plus gas, with an electron, plus Cl gas, then down by 349 to gaseous Na plus and Cl minus ions. A downward arrow of minus 787 labelled lattice energy reaches the solid NaCl at minus 411. A long arrow from the elements straight down to the solid is labelled delta H f, minus 411.
Born-Haber cycle for NaCl, drawn to scale. The staircase and the long arrow are two routes between the same two levels.

Adding the steps up shows where each level sits, measured from the elements:

Level Species Enthalpy / kJ mol⁻¹
0 Na(s)+12 ClX2(g)\ce{Na(s) + 1/2 Cl2(g)} 0
1 Na(g)+12 ClX2(g)\ce{Na(g) + 1/2 Cl2(g)} +107
2 Na(g)+Cl(g)\ce{Na(g) + Cl(g)} +229
3 NaX+(g)+eX−+Cl(g)\ce{Na+(g) + e- + Cl(g)} +725
4 NaX+(g)+ClX−(g)\ce{Na+(g) + Cl-(g)} +376
5 NaCl(s)\ce{NaCl(s)} −411

The final step goes from +376 to −411, a change of −787 kJ mol⁻¹. That is the lattice energy, found without measuring it.

MethodDrawing a Born-Haber cycle

  1. Put the elements in their standard states at the bottom left, as the zero level.
  2. Atomise each element, with the amounts the formula needs: one mole of atoms for each atom of the formula. Two chlorine atoms need 2×ΔatH2 \times \Delta_\text{at}H(Cl).
  3. Ionise the metal, with one ionisation energy for each electron it loses. Keep the electrons in the species label until they are taken up.
  4. Add the electron affinity of the non-metal, once per electron gained.
  5. Join the gaseous ions to the solid with the lattice energy, the last downward arrow.
  6. Draw ΔfH\Delta_\text{f}H from the elements to the solid. Label every arrow with its name and value, and every level with its species and state symbols.

Check yourself

  1. Write the five steps, with equations and state symbols, of a Born-Haber cycle for potassium chloride, KCl. Use ΔfH\Delta_\text{f}H(KCl) = −437 kJ mol⁻¹ and give the value of each known step: ΔatH\Delta_\text{at}H(K) = +89, ΔatH\Delta_\text{at}H(Cl) = +122, first ionisation energy of K = +419, first electron affinity of Cl = −349 (all kJ mol⁻¹).
  2. For the KCl cycle, find the enthalpy of each level from the elements. Which step is the lattice energy?
  3. In the NaCl cycle, which steps are endothermic and which are exothermic? How does each appear on the diagram?
  4. Why is the electron included in the label at the third level, NaX+(g)+eX−+Cl(g)\ce{Na+(g) + e- + Cl(g)}?
  5. Why can the lattice energy not be measured directly?

Answers cover these until you have tried the questions

  1. (1) K(s)→K(g)\ce{K(s) -> K(g)}, +89. (2) 12 ClX2(g)→Cl(g)\ce{1/2 Cl2(g) -> Cl(g)}, +122. (3) K(g)→KX+(g)+eX−\ce{K(g) -> K+(g) + e-}, +419. (4) Cl(g)+eX−→ClX−(g)\ce{Cl(g) + e- -> Cl-(g)}, −349. (5) KX+(g)+ClX−(g)→KCl(s)\ce{K+(g) + Cl-(g) -> KCl(s)}, the lattice energy, unknown.
  2. Levels from the elements at 0: +89, then 89 + 122 = +211, then 211 + 419 = +630, then 630 − 349 = +281 (KX+(g)+ClX−(g)\ce{K+(g) + Cl-(g)}). The solid KCl is at −437. Step 5 goes from +281 to −437, which is −718 kJ mol⁻¹: that is the lattice energy.
  3. Steps 1, 2 and 3 (atomisation of each element, and ionisation of sodium) are endothermic and are drawn as arrows going up. Step 4 (the electron affinity of chlorine) and step 5 (the lattice energy) are exothermic and are drawn as arrows going down.
  4. The electron removed from sodium still exists until chlorine accepts it in the next step. Including it keeps the species and the charges balanced at every level.
  5. It would need separate gaseous ions to be brought together, which cannot be done in a calorimeter. It is found indirectly using Hess's law, from the formation enthalpy and the other, measurable, steps.